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I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen. Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\...

Let $\Phi (z,t)$ be a polynomial given by $$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$ Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation $$ \...